There’s no core magic items that match exactly what you’re asking for. However, 5E has optional rules for crafting magic items, and a little tiny bit of guidance for DMs on how to design them. That said, you’re firmly in homebrew territory if you go this route.
A tiny bit of guidance…
Now, as for what that would look like, the healing spell alone–you can get away with Cure Wounds a 1st level spell. Per this table in the DMG, a once per day 1st level spell could be a Common magic item.
However, the ‘auto-cast’ feature of it is, essentially, a Contingency spell. “If I hit 0hp, cast Cure Wounds on self.” And that’s a 6th level spell…which would (again, per that very vague table) put us up in Rare territory.
If you go with the DMG’s rules for this crafting, you’re looking at a 5,000gp expenditure and 200 man-days of effort.
If you go with XGtE’s rules, you’re looking at 10 workweeks of effort and 2,000gp, plus some ingredient pulled off a CR 9-12 creature.
This is just spitballing the effect here…homebrewing magic items is a very touchy affair and it’s really left up to the DM on what it’s going to look like, what its rarity would be, and how much it’s going to take to create it.
But, for reference, the only other “Auto Healing” item I can find in the core rules is the Ring of Regeneration, which is a Very Rare item. (though it heals constantly, not limited per day)
Periapt of Wound Closure: Uncommon
While you wear this pendant, you stabilize whenever you are dying at the start of your turn. In addition, whenever you roll a Hit Die to regain hit points, double the number of hit points it restores.
It doesn’t directly heal you, but it means that if you’re at 0HP, you don’t have to worry about death saves unless enemies start wailing on you while you’re down. And it doubles the value of your Hit Dice…so a character who gets beat up a lot can recover from it, without magical healing, more effectively.
Were I DMing for this player, this is what I’d suggest they go for.